The second issue is the non-uniform convergence of the estimate when
\(k\)  is increased, which limits the regions in which statistical errors
can be defined. In order to define a statistical error, one normally
assumes that the distribution of the sum over the \(N_{b}\)  values
converges to a Gaussian distribution around its mean. Statistical
errors are then defined based on the standard deviation of the distribution.
For of sum of exponentials of random variables, like in our case,
this is true only on half of the convergence region of the estimator
[1]}. For \(k_{c}^{-}(\tau _{b},N_{b})/2<k<k_{c}^{+}(\tau _{b},N_{b})/2\) ,
the estimators converge, they are Gaussian-distributed, and statistical
errors can be computed from their empirical variance. For \(k_{c}^{-}(\tau _{b},N_{b})<k<k_{c}^{-}(\tau _{b},N_{b})/2\) 
or \(k_{c}^{+}(\tau _{b},N_{b})/2<k<k_{c}^{+}(\tau _{b},N_{b})\) , the
estimators converge, but they are not Gaussian-distributed, and their
statistical error cannot be determined from the empirical variance.
The estimates in these regions therefore cannot be properly analyzed.
In the inner convergence region, the statistical error on the estimate
of \(G(k)\)  is computed as the standard deviation of the sample of
values involved in the sum replacing the expectation value. The statistical
errors on the estimates of \(\lambda (k)\)  and \(I(a)\)  can then be computed
by error propagation, as described in Appendix .
